The concept of displacement structure has been used to solve several problems connected with Toeplitz matrices and with matrices obtained in some way from Toeplitz matrices (e.g., by combinations of multiplication, inversion, and factorization). Matrices of the latter type will be called Toeplitz-derived (or Toeplitz-like, close-to-Toeplitz). This paper introduces a generalized definition of displacement for block-Toeplitz and Toeplitz-block arrays. It will turn out that Toeplitz-derived matrices are perhaps best regarded as particular Schur complements obtained from suitably defined block matrices. The new displacement structure is used to obtain a generalized Schur algorithm for fast triangular and orthogonal factorizations of all such matrices and well-structured fast solutions of the corresponding exact and overdetermined systems of linear equations. Furthermore, this approach gives a natural generalization of the so-called Gohberg-Semencul formulas for Toeplitz-derived matrices.